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Stability of graphical tori with almost nonnegative scalar curvature

By works of Schoen-Yau and Gromov-Lawson any Riemannian manifold with nonnegative scalar curvature and diffeomorphic to a torus is isometric to a flat torus. Gromov conjectured subconvergence of tori with respect to a weak Sobolev type metric when the scalar curvature goes to $0$. We prove flat and intrinsic flat subconvergence to a flat torus for noncollapsing sequences of $3$-dimensional tori $M_j$ that can be realized as graphs of certain functions defined over flat tori satisfying a uniform upper diameter bound and scalar curvature bounds of the form $R_{g_{M_j}} \geq -1/j$. We also show that the volume of the manifolds of the convergent subsequence converges to the volume of the limit space. We do so adapting results of Huang-Lee, Huang-Lee-Sormani and Allen-Perales-Sormani. Furthermore, our results also hold when the condition on the scalar curvature of a torus $(M, g_M)$ is replaced by a bound on the quantity $-\int_T \min\{R_{g_M},0\} d{\mbox{vol}_{g_T}}$, where $M=\mbox{graph}(f)$, $f: T \to \mathbb R$ and $(T,g_T)$ is a flat torus. Using arguments developed by Alaee, McCormick and the first named author after this work was completed, our results hold for dimensions $n \geq 4$ as well.